Let $F$ be a field of characteristic different from 2,
$\psi$ a quadratic $F$-form of dimension $\geq5$, and $D$ a
central simple $F$-algebra of exponent 2.
We denote by $F(\psi,D)$ the function field of the product
$X_\psi\times X_D$, where $X_\psi$ is the projective quadric
determined by $\psi$ and $X_D$ is the Severi-Brauer
variety determined by $D$.
We compute the relative Galois cohomology group
$H^3(F(\psi,D)/F,\Z/2\Z)$ under the
assumption that the index of $D$ goes down when extending
the scalars to $F(\psi)$.
Using this, we give
a new, shorter proof
of the theorem [23, Th. 1]
originally proved by A. Laghribi,
and
a new, shorter, and more elementary proof of
the assertion [2, Cor. 9.2]
originally proved by
H. Esnault, B. Kahn, M. Levine, and E. Viehweg.